Any fact about these would be of interest. Has anyone seen an interesting structure constructed from a monad and comonad based on a single identity functor?

I am working on a very small example just so that I can do all the calculations. In particular, suppose I have a category with one object that's a set with two elements. The morphisms are all endo-functions of this set. Now take the identity endo-functor and use it to generate a monad and comonad. Next, find the (co)algebra for these. Are these completely trivial?